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Jackson theorems for Erdős weights in \(L_p\) \((0<p\leq \infty)\). (English) Zbl 0915.41013
This paper is a the first part of a work on Jackson estimates for the approximation by polynomials, of functions in weighted \(L_p\), \(0<p\leq\infty\), where the weights are Erdős type weights. The second part contains inverse theorems and was written by the first author. For some reason the latter appeared earlier, in [J. Approximation Theory 93, No. 3, 349-398 (1998; Zbl 0907.41010]. In this paper the authors establish the Jackson estimates by introducing new moduli of smoothness, based on the Erdős weights and contain two parts. One which is given by the weighted norm of the differences of the function on a suitably defined finite interval, and the other which measures the best of approximation by polynomials of low degrees, on the infinite part of the interval. The above mentioned second paper shows that these moduli are indeed the proper ones. The proofs are quite technical. The authors first approximate the function by piecewise polynomials of low degree as best as possible (as guaranteed by Whitney), and then replace the piecewise polynomials by polynomials of the much higher degree.

41A30 Approximation by other special function classes
41A10 Approximation by polynomials
Full Text: DOI
[1] Brudnyi, Yu.A., Approximation of functions by algebraic polynomials, Izv. akad. nauk SSSR ser. math., 32, 780-787, (1968) · Zbl 0165.38702
[2] Clunie, J.; Kövari, T., On integral functions having prescribed asymptotic growth, II, Canad. J. math., 20, 7-20, (1968) · Zbl 0164.08602
[3] Damelin, S.B., Converse and smoothness theorems for erdo&#x030B;s weights in lp(0<p, J. approx. theory, 93, 349-398, (1998) · Zbl 0907.41010
[4] DeVore, R.A.; Popov, V.A., Interpolation of Besov spaces, Trans. amer. math. soc., 305, 397-414, (1988) · Zbl 0646.46030
[5] DeVore, R.A.; Leviatan, D.; Yu, X.M., Polynomial approximation inL_p(0<p, Constr. approx., 8, 187-201, (1992)
[6] Ditzian, Z.; Lubinsky, D.S., Jackson and smoothness theorems for freud weights inL_p, 0<p, Constr. approx., 13, 99-152, (1997) · Zbl 0867.41010
[7] Ditzian, Z.; Totik, V., Moduli of smoothness, Springer series in computational mathematics, (1987), Springer-Verlag Berlin · Zbl 0666.41001
[8] Freud, G., Orthogonal polynomials, (1971), Pergamon/Akademiai Kiado Budapest · Zbl 0226.33014
[9] von Golitschek, M.; Lorentz, G.G.; Makovoz, Y., Asymptotics of weighted polynomials, (), 431-451 · Zbl 0806.41008
[10] Jansche, S.; Stens, R.L., Best weighted polynomial approximation on the real line: A functional analytic pproach, J. comput. appl. math., 40, 199-213, (1992) · Zbl 0756.41022
[11] D. Leviatan, X. M. Yu, Shape preserving approximation by polynomials inL_p
[12] Levin, A.L.; Lubinsky, D.S., Christoffel functions, orthogonal polynomials, and Nevai’s conjecture for freud weights, Constr. approx., 8, 463-535, (1992) · Zbl 0762.41011
[13] Levin, A.L.; Lubinsky, D.S.; Mthembu, T.Z., Christoffel functions and orthogonal polynomials for erdo&#x030B;s weights on (−∞,∞), Rend. mat. appl., 14, 199-289, (1994) · Zbl 0816.42015
[14] Lubinsky, D.S., Strong asymptotics for extremal errors and polynomials associated with rdo&#x030B;s type weights, Pitman research notes in mathematics, (1989), Longman Harlow · Zbl 0743.41001
[15] Lubinsky, D.S., An update on orthogonal polynomials and weighted approximation on the real line, Acta appl. math., 33, 121-164, (1993) · Zbl 0799.42013
[16] Lubinsky, D.S., Ideas of weighted polynomial approximation on (−∞,∞), Approximation and interpolation, 371-396, (1995) · Zbl 1137.41307
[17] Lubinsky, D.S.; Mthembu, T.Z., Orthogonal expansions and the error of weighted polynomial approximation for rdo&#x030B;s weights, Numer. funct. anal. and optimiz., 13, 327-347, (1992) · Zbl 0767.41012
[18] Mhaskar, H.N.; Saff, E.B., Extremal problems for polynomials with exponential weights, Trans. amer. math. soc., 285, 203-234, (1984) · Zbl 0546.41014
[19] Mhaskar, H.N.; Saff, E.B., Where does the sup-norm of a weighted polynomial live?, Constr. approx., 1, 71-91, (1985) · Zbl 0582.41009
[20] Mhaskar, H.N.; Saff, E.B., Where does theL_pnorm of a weighted polynomial live?, Trans. amer. math. soc., 303, 109-124, (1987) · Zbl 0636.41008
[21] Nevai, P., G. freud, orthogonal polynomials and Christoffel functions: A case study, J. approx. theory, 48, 3-167, (1986) · Zbl 0606.42020
[22] Nevai, P., Orthogonal polynomials, theory and practice, NATO ASI series, 294, (1990), Kluwer Academic Dordrecht
[23] Oswald, P., Ungleichungen vom Jackson-typ für die algebraische beste approximation inL_p, J. approx. theory, 23, 113-136, (1978) · Zbl 0379.41015
[24] Petrushev, P.P.; Popov, V., Rational approximation of real functions, (1987), Cambridge Univ. Press Cambridge · Zbl 0644.41010
[25] Saff, E.B.; Totik, V., Logarithmic potentials with external fields, (1997), Springer-Verlag New York/Berlin · Zbl 0881.31001
[26] Totik, V., Weighted approximation with varying weight, Lecture notes in mathematics, (1994), Springer-Verlag Berlin · Zbl 0808.41001
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